Exponential Stability of Periodic Solutions in Inertial Neural Networks with Unbounded Delay
In this paper, the exponential stability of periodic solutions in inertial neural networks with unbounded delay are investigated. First, using variable substitution the system is transformed to first order differential equation. Second, by the fixed-point theorem and constructing suitable Lyapunov function, some sufficient conditions guaranteeing the existence and exponential stability of periodic solutions of the system are obtained. Finally, two examples are given to illustrate the effectiveness of the results.
<p>[1] L.O. Chua, L. Yang, Cellular neural networks: Theory, IEEE Trans.
Circuits Syst. 35 (1988) 1257-1272.
[2] L.O. Chua , L. Yang, Cellular neural networks: Applications, IEEE Trans.
Circuits Syst. 35 (1988) 1273-1290.
[3] L.O. Chua, CNN: A Paradigm for Complexity, World Scientific, Singapore,
1998.
[4] T. Fariaa, J. Oliveirab, General criteria for asymptotic and exponential
stabilities of neural network models with unbounded delays, Applied
Mathematics and Computation 217(23)(2011) 9646-9658.
[5] W. Zhaoa, A. Yan, Stability analysis of neural networks with both variable
and unbounded delays, Chaos, Solitons , Fractals, 40( 2009) 697-707.
[6] J.Y. Zhang, Y. Suda, T. Iwas, Absolutely exponential stability of a class
of neural networks with unbounded delay, Neural Networks, 17(3)( 2004)
391-397.
[7] Z.G. Zeng, J.Wang, X.X.Liao, Global asymptotic stability and global
exponential stability of neural networks with unbounded time-varying
delays, IEEE Trans Circuits Syst II: Express Briefs, 52(3)(2005) 168-73.
[8] Z.G. Zeng , X.X. Liao, Global stability for neural networks with unbounded
time varying delays, Acta Mathematica Scientia, 25A(5)(2005)
621-626.
[9] Y.Zhang, P.A. Heng, K.S.Leung, Convergence analysis of cellular neural
networks with unbounded delay, IEEE Trans Circuits Syst I: Fundam
Theor Appl, 48(6)( 2001) 680-687.
[10] Q. Zhang, X.P.Wei, J. Xu, Global exponential stability for nonautonomous
cellular neural networks with unbounded delays, em Chaos,
Solitons Fractals, 39(3)( 2009) 1144-1151.
[11] W.R.Zhao, A.Z.Yan, Stability analysis of neural networks with both
variable and unbounded delays, Chaos, Solitons Fractals, 40(2)( 2009)
697-707.
[12] J.Zhang, Absolutely stability analysis in cellular neural networks with
variable delays and unbounded delay, Comput Math. Appl. 47(2004) 183-
194.
[13] Z. Zeng, J. Wang, X. Liao, Global asymptotic stability and global
exponential stability of neural networks with unbounded time-varying
delays, IEEE Transactions on Circuits Systems II , 52 (3)( 2005) 168-
173.
[14] J. Zhang, Y. Suda, T. Iwasa, Absolutely exponential stability of a class
of neural networks with unbounded delay, Neural Networks, 17( 2004)
391-397.
[15] K.L. Badcock, R.M. Westervelt, Dynamics of simple electronic neural
networks, Physica D. 28( 1987) 305-316.
[16] J. Tani, Proposal of chaotic steepest descent method for neural networks
and analysis of their dynamics, Electron. Comm. Japan, 75(4)(1992) 62-
70.
[17] J. Tani, M. Fujita, Coupling of memory search and mental rotation by a
nonequilibrium dynamics neural network, IEICE Trans. Fund. Electron.,
Commun. Comput. Sci. E75-A (5)(1992) 578-585.
[18] J. Tani, Model-based learning for mobile robot navigation from the
dynamical systems perspective, IEEE Trans. Systems Man Cybernet 26
(3)( 1996) 421-436.
[19] C.G.Li, G.R.Chen, X.F. Liao, J.B. Yu, Hopf bifurcation and chaos in a
single inertial neuron model with time delay, Eur.Phys. J. B, 41( 2004)
337-343.
[20] Q. Liu, X.F. Liao, G.Y. Wang, Y. Wu, Research for Hopf bifurcation
of an inertial two-neuron system with time delay, in: IEEE Proceeding
GRC, (2006) 420-423.
[21] Q. Liu, X.F. Liao, D.G. Yang, S.T. Guo, The research for Hopf
bifurcation in a single inertial neuron model with external forcing, in:
IEEE Proceeding GRC, ( 2007) 528-533.
[22] Q. Liu, X. Liao, Y. Liu, S. Zhou , S. Guo, Dynamics of an inertial
two-neuron system with time delay, Nonlinear Dyn. 58( 2009) 573-609.
[23] W. Diek, W.C. Wheeler, Schieve.Stability and chaos in an inertial twoneuron
system, Physica D, 105 (1997) 267-284.
[24] Q. Liu , X.F. Liao, S.T. Guo, Y. Wu, Stability of bifurcating periodic
solutions for a single delayed inertial neuron model under periodic
excitation, Nonlinear Analysis: Real World Applications, 10(2009) 2384-
2395.
[25] Y.Q. Ke, C.F. Miao, Stability analysis of BAM neural networks with
inertial term and time delay, WSEAS. Transactions on System , 10(12)(
2011) 425-438.
[26] Y.Q. Ke, C.F. Miao, Stability and existence of periodic solutions in
inertial BAM neural networks with time delay, Neural computing and
Applications, (2012, DOI: 10.1007/s00521-012-1037-8, Online First, 15
July 2012).</p>
<p>[1] L.O. Chua, L. Yang, Cellular neural networks: Theory, IEEE Trans.
Circuits Syst. 35 (1988) 1257-1272.
[2] L.O. Chua , L. Yang, Cellular neural networks: Applications, IEEE Trans.
Circuits Syst. 35 (1988) 1273-1290.
[3] L.O. Chua, CNN: A Paradigm for Complexity, World Scientific, Singapore,
1998.
[4] T. Fariaa, J. Oliveirab, General criteria for asymptotic and exponential
stabilities of neural network models with unbounded delays, Applied
Mathematics and Computation 217(23)(2011) 9646-9658.
[5] W. Zhaoa, A. Yan, Stability analysis of neural networks with both variable
and unbounded delays, Chaos, Solitons , Fractals, 40( 2009) 697-707.
[6] J.Y. Zhang, Y. Suda, T. Iwas, Absolutely exponential stability of a class
of neural networks with unbounded delay, Neural Networks, 17(3)( 2004)
391-397.
[7] Z.G. Zeng, J.Wang, X.X.Liao, Global asymptotic stability and global
exponential stability of neural networks with unbounded time-varying
delays, IEEE Trans Circuits Syst II: Express Briefs, 52(3)(2005) 168-73.
[8] Z.G. Zeng , X.X. Liao, Global stability for neural networks with unbounded
time varying delays, Acta Mathematica Scientia, 25A(5)(2005)
621-626.
[9] Y.Zhang, P.A. Heng, K.S.Leung, Convergence analysis of cellular neural
networks with unbounded delay, IEEE Trans Circuits Syst I: Fundam
Theor Appl, 48(6)( 2001) 680-687.
[10] Q. Zhang, X.P.Wei, J. Xu, Global exponential stability for nonautonomous
cellular neural networks with unbounded delays, em Chaos,
Solitons Fractals, 39(3)( 2009) 1144-1151.
[11] W.R.Zhao, A.Z.Yan, Stability analysis of neural networks with both
variable and unbounded delays, Chaos, Solitons Fractals, 40(2)( 2009)
697-707.
[12] J.Zhang, Absolutely stability analysis in cellular neural networks with
variable delays and unbounded delay, Comput Math. Appl. 47(2004) 183-
194.
[13] Z. Zeng, J. Wang, X. Liao, Global asymptotic stability and global
exponential stability of neural networks with unbounded time-varying
delays, IEEE Transactions on Circuits Systems II , 52 (3)( 2005) 168-
173.
[14] J. Zhang, Y. Suda, T. Iwasa, Absolutely exponential stability of a class
of neural networks with unbounded delay, Neural Networks, 17( 2004)
391-397.
[15] K.L. Badcock, R.M. Westervelt, Dynamics of simple electronic neural
networks, Physica D. 28( 1987) 305-316.
[16] J. Tani, Proposal of chaotic steepest descent method for neural networks
and analysis of their dynamics, Electron. Comm. Japan, 75(4)(1992) 62-
70.
[17] J. Tani, M. Fujita, Coupling of memory search and mental rotation by a
nonequilibrium dynamics neural network, IEICE Trans. Fund. Electron.,
Commun. Comput. Sci. E75-A (5)(1992) 578-585.
[18] J. Tani, Model-based learning for mobile robot navigation from the
dynamical systems perspective, IEEE Trans. Systems Man Cybernet 26
(3)( 1996) 421-436.
[19] C.G.Li, G.R.Chen, X.F. Liao, J.B. Yu, Hopf bifurcation and chaos in a
single inertial neuron model with time delay, Eur.Phys. J. B, 41( 2004)
337-343.
[20] Q. Liu, X.F. Liao, G.Y. Wang, Y. Wu, Research for Hopf bifurcation
of an inertial two-neuron system with time delay, in: IEEE Proceeding
GRC, (2006) 420-423.
[21] Q. Liu, X.F. Liao, D.G. Yang, S.T. Guo, The research for Hopf
bifurcation in a single inertial neuron model with external forcing, in:
IEEE Proceeding GRC, ( 2007) 528-533.
[22] Q. Liu, X. Liao, Y. Liu, S. Zhou , S. Guo, Dynamics of an inertial
two-neuron system with time delay, Nonlinear Dyn. 58( 2009) 573-609.
[23] W. Diek, W.C. Wheeler, Schieve.Stability and chaos in an inertial twoneuron
system, Physica D, 105 (1997) 267-284.
[24] Q. Liu , X.F. Liao, S.T. Guo, Y. Wu, Stability of bifurcating periodic
solutions for a single delayed inertial neuron model under periodic
excitation, Nonlinear Analysis: Real World Applications, 10(2009) 2384-
2395.
[25] Y.Q. Ke, C.F. Miao, Stability analysis of BAM neural networks with
inertial term and time delay, WSEAS. Transactions on System , 10(12)(
2011) 425-438.
[26] Y.Q. Ke, C.F. Miao, Stability and existence of periodic solutions in
inertial BAM neural networks with time delay, Neural computing and
Applications, (2012, DOI: 10.1007/s00521-012-1037-8, Online First, 15
July 2012).</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65299", author = "Yunquan Ke and Chunfang Miao", title = "Exponential Stability of Periodic Solutions in Inertial Neural Networks with Unbounded Delay", abstract = "In this paper, the exponential stability of periodic solutions in inertial neural networks with unbounded delay are investigated. First, using variable substitution the system is transformed to first order differential equation. Second, by the fixed-point theorem and constructing suitable Lyapunov function, some sufficient conditions guaranteeing the existence and exponential stability of periodic solutions of the system are obtained. Finally, two examples are given to illustrate the effectiveness of the results.
", keywords = "Inertial neural networks, unbounded delay, fixed-point theorem, Lyapunov function, periodic solutions, exponential stability.", volume = "7", number = "3", pages = "462-10", }
